Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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Continuous maps in from Banach space to $\ell ^\infty$

Let $X$ be a Banach space. Prove that a linear map $M\colon X\mapsto \ell^p, \; p\geqslant 1$ is continuous iff for every sequence $(x_k)$ that converges in $X$ to $x \in X$, we have that the $n$-th ...
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171 views

Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
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80 views

Inverse of a certain differential operator (resolvent)

I am doing a research on a certain type of operator, and in the course of it I need to determine the following: Given the operator $D$ below, and identity operator $I$, $$ D=\begin{pmatrix} ...
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84 views

Uniform limit of finite-rank operators with the same rank.

Let $\{T_n\in\mathcal{B}(X)\,|\,\text{rank}(T_n)=R\,\}^{\infty}_{n=1}$ is a sequence of linear bounded finite-rank operators on a Banach space with the same rank $R$. Let it converge uniformly to an ...
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84 views

Compactness of operator $M: C([0,1]) \rightarrow C([0,1])$

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Is this operator compact? I have trouble using limit in operator norm of compact operator, or cauchy ...
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78 views

Range of operator $ Mf(x) = f(x/2), \;\; x\in[0,1]$

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Prove that the range of $I-M$ does not contain nonzero constant functions, but it contains all functions ...
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103 views

bounded operator between continuous functions

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Show that $M$ is bounded and that its spectrum is containd in the closed unit disc $\{ \lambda \in \mathbb{C} ...
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60 views

Subspace in $I-T$ for bounded linear maps

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map.Show that the range of $I - T$ contains the subspace $$Y_T = \{x \in X: \limsup_{n\rightarrow \infty} n^2\|T^nx\| < ...
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239 views

Bounded linear maps in Banach spaces

Let $X$ be a Banach space and let $M: X \rightarrow X$ be a linear map. Prov that M is bounded iff there exists a set $S \subset X'$, dense in X', such that for each $\ell \in S$ the functional $m_l$ ...
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87 views

Closed extensions in the weak* topology

Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm. and let $\ell_0(x) = \lim_{n \rightarrow \infty} x_n$, for convergent sequences. Show that the sett L consisting of all ...
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409 views

Show that a finite-dimensional Banach space has a bijective compact operator

It is clear that if $ T: X \rightarrow X $ is a bijective compact operator, where $ X $ is a Banach space, then $ \dim(\text{Range}(T)) = \dim(X) $, which implies that $ \dim(X) $ must be $ < ...
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175 views

Spectrum in an separable Hilbert space

Let $H$ be a separable Hilbert space with orthonormal basis $\{e_i\}$. Let $(c_n)$ be a bounded sequence of complex numbers and consider the bounded linear operator $T$ on $H$ defined by $$Tx = ...
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265 views

Compute the spectrum for a operator

Find the spectrum of the operator $$ \begin{split} A & \colon C[0,1] \rightarrow C[0,1] \\ & f \mapsto (Af)(x) := f(x) + \int_0^x f(t)dt \end{split} $$ P.S.: I know the spectrum ...
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84 views

Banach limit and its commutative counterpart, what do they tell us?

A Banach limit is a continuous linear functional $\Lambda$ on $\ell^{\infty}(\mathbb{N})$ satisfying: $\|\Lambda\|=\Lambda(1,1,1,\cdots)=1$; and ...
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40 views

Skew-symmetric unitary on $\mathcal{B(H)}$

We know that there exists skew-symmetric unitary on $\mathcal{B(H)}$ when $\mathcal{H}$ is of even dimensions. In particular for $\mathcal{H}=\mathbb{C}^2$, any such matrix is scalar multiple of Pauli ...
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252 views

boundedness of an operator

Define $T: L^2(\mathbb{R})\to L^2(\mathbb{R})$ by $(Tf)(x)=\int_{\mathbb{R}}\frac{f(y)}{1+|x|+|y|}dy$. Is this operator bounded? If it is, then is it also compact? I got stuck in simply applying ...
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88 views

Differential Operator on $L_{2}$ problem

I am working on a problem from a textbook and have run into difficulties on this specific question. Any assistance will be appreciated, Consider the partial differential equation, $\frac{\partial ...
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74 views

Toeplitz Operator question

Let $\chi_1$ be the map on the unit circle defined by $\chi_1(e^{it})=e^{it}$. Let $T_{\chi_1}$ be the corresponding Toeplitz operator. Consider the map $T_{\chi_1}^* T_{\chi_1}- T_{\chi_1} ...
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76 views

Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
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Hilbert transform and Hilbert matrix

The Hilbert matrix is \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt] ...
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78 views

Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out. Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ ...
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44 views

Using “adjunction” to refer to the act of taking adjoints of operators

I have an especially flabby terminology question. How acceptable is it, in your opinion, to use the word "adjunction" to refer to the process of taking adjoints of operators on a Hilbert space? ...
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215 views

Operators bounded below

Can one give me an easy example of an operator $T$ on a Banach space which is injective and has closed range and such that $\|T^2\|\neq \|T\|^2$?
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51 views

Operator defined via a sequence of weights

Let the linear operator $T:l^2\rightarrow l^2$ be defined by $y=Tx$ where $x=\{\xi_j\}$, $y=\{\eta_j\}$, and $\eta_j = \alpha_j \xi_j$, where $\{\alpha_j\}$ is a dense sequence in $[0,1]$. Does ...
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278 views

Every Hilbert-Schmidt is an integral operator?

Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation} ...
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486 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
3
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270 views

cyclic vector exists for symmetric operator iff there no repeated eigenvalues

Considering a symmetric operator $A$ acting on a finite dimensional Hilbert space $H$, we say $x\in H$ is a cyclic vector for $A$ if the set of finite linear combinations of $\{A^n x:n=0,1,2,...\}$ is ...
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237 views

Compact integral and multiplication operator in Banach spaces

Let $ A\colon C[0,1] \to C[0,1] $ $$ A(x)(t) = f(t)x(t) + \int_0^t x(s)ds,\quad f \in C[0,1]: f(1) \neq 0, \forall t \in [0,1] $$ Is $A$ a compact operator or not?
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Operators with eigenvalue $\{0,1\}$ that is not projection

Show that there are linear operators T on the Hilbert space H what are not orthogonal projections, but their spectrum consists of the eigenvalues $\{0,1\}.$ I can not come up with an counterexample, ...
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75 views

Power series of bounded linear maps

Given a Banach space $X$ and a bounded linear map $T:X\rightarrow X$ we define $$e^T = I + \sum_{n\geq1}\frac{T^n}{n!}$$ Show that if $e^T$ is compact then dim $X<\infty$. I have showed before ...
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68 views

Spectral raius for linear compact maps

Prove or disprove the following assertions for a linear map $C$ from a Banach space $X$ into itself: a) If C is compact then its spectral radius equals the maximum of the absolute value of $C$ Im ...
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310 views

Operators on $C([0,1])$ that is compact or not.

For $f\in C([0,1])$ set $$Hf(x) = \frac{1}{x}\int_0^x f(t)dt.$$ a) Show that $H$ is a bounded operator from $C([0,1])$ into itself which is not compact. b) From a) it follows that $H$ induces a ...
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147 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
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56 views

Approximating bounded operators in Hilbert space

Let $H$ be a separable Hilbert space, show that every bounded operator from H to itself can be approximated in the strong operator topology by a sequence of finite rank operators. I know we can find ...
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165 views

Graph of symmetric linear map is closed

A homework problem: Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$). Show that $S$ is bounded. My attempt: I'd ...
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56 views

Characterize compact sequences for a linear map.

Given a bounded sequence $\pi = (\lambda_n)$ in $\mathbb{C}$ consider the continuous linear map $M_\pi:\ell^2\rightarrow \ell^2$ defined by $$M_\pi(x_n) = (\pi_nx_n)$$ a) determine the spectrum. b) ...
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194 views

The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
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143 views

The convergence of the adjoint operator

If a sequence of operator $A_n$ converges in norm to $A$, i.e. $\lim \lVert A_n-A\rVert=0$)where $A_n$ and $A\in B(H)$ ($H$ is the Hilbert space). Is it true that $A_n^*$ converges in norm to $A^*$?
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168 views

Normal operators in Hilbert spaces

Let $H$ be a separable Hilbert space and let $T:H\to H$ be a continues linear map such that there exists an orthonormal basis of $H$ that consists of the eigenvectors of $T$. Show that $T$ is normal. ...
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143 views

Determine the operator T in a Hilbert space

Let $H$ be a Hilbert space and let $\{e_n, n \geq 1\}$ be an orthonormal basis for $H$. a) Determine the operator $T\in B(H)$ that satisfies $$ Te_1 = 0,\; Te_n = \frac{1}{n}e_{n-1}, n ...
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440 views

Projection operator in Hilbert spaces

Let T be a bounded operator on the Hilbert space H with the property that $T^*(T-I)= 0$. Show that T is an orthogonal projection. Im not really sure how to show that an operator is an orthogonal ...
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Convergence of a series in the $C^\infty$ topology

I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators". The motivating problem for this is to find an approximate kernel ...
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86 views

Show that $(x_n)$ is in $\ell^2$

Let $x = (x_n)$ be a sequence of complex numbers with the property that for every $y = (y_n) \in \ell^2$ we have that the sequence $(S_N(y))_{N\geq1}$ with $$S_N(y) =\sum_{n=1}^N x_ny_n $$ converges. ...
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331 views

If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm.

If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm. I can show that $A$ would be positive and thus have a ...
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160 views

If a map $C:X\rightarrow U$ maps every weakly convergent sequence into strongly convergent

A Linear map between Banach spaces $C:X\rightarrow U$ is compact if it maps if the closure of the image of the unit ball is precompact in U. If a map $C:X\rightarrow U$ maps every weakly convergent ...
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Does $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for bounded operators on Hilbert space?

If $A$ is a bounded linear operator on a Hilbert space $H$ is it true that $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for all $x\in H$? If not, can we at least establish inequality in one ...
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82 views

Unbounded sets in infinite-dimensional normed spaces.

Let $X$ be an infinte-dimensional normed space. Let $\ell_1,\ldots, \ell_n$ be continuous linear functionals on $X$ and consider the set $$U = \{x\in X : |\ell_j(x)| < 1,\;\; 1\leq j \leq n\}.$$ ...
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70 views

Collection of linear functions

Let $X$ be a Banach space. Let $\{Y_\alpha\}_\alpha$ be normed spaces. Let $\{T_\alpha:X\rightarrow Y_\alpha\}_\alpha$ be an infinite collection of bounded linear functions. Is there a way to create ...
2
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1answer
291 views

Transpose of Volterra operator

I want to find the transpose of the Volterra operator $$Vf(x) = \int_0^x f(t)dt, \;\; x\in(0,1)$$ acting in $V:L^2(0,1) \rightarrow V:L^2(0,1) $. The transpose is defined as $\textbf{M}':U'\rightarrow ...
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65 views

Compactness of multiplication operator [duplicate]

Possible Duplicate: Compactness of Multiplication Operator on $L^2$ Let $u: \mathbb{R}\rightarrow \mathbb{C}$ be a bounded continuous function. Show that the multiplication operator $M_u$ ...